A general relativity formulation for the Doplicher-Fredenhagen-Roberts noncommutative space-time

Abstract

The Doplicher, Fredenhagen and Roberts (DFR) noncommutative (NC) formalism was proposed in a curved space-time. In the DFR approach, the NC parameters are promoted to a set of space-time coordinates. Consequently, the field theory defined on this space contains extra dimensions. We propose a metric containing the new coordinates to explain the length measurements of this extended space. We promote the Minkowski part of this extended metric to a metric defined in a curved manifold. Thus, we can build up a NC gravitation model with extra dimensions. Based on the curved metric, we propose an Einstein-Hilbert action to describe the NC general relativity model on the DFR space. It is a new result in the literature since there is not a general relativity formalism in the DFR approach. We also introduce the NC version of the general relativistic field equations. As an application of the formalism, the weak-field approximation is considered in the field equations to generate the wave equations for the graviton propagation in DFR space.